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A MODEL OF TOUGHENING EFFECTSIN WHISKER-REINFORCED COMPOSITES

PNL-SA--16936

R. G. Hoaglandl and C. H. Henager, Jr.2DE92 000226

1Department of Mechanical and Materials EngineeringWashington Sta:e University

Pullman, Washington 99164-2920_acific Northwest LaboratoryRichland, Washington 99352

Abstract

A numerical approach is presented that lends itself to modeling the screening or

antiscreening effects due solely to modulus differences of a discrete array of whiskers in an

elastic matrix. The method is applied to single whiskers, aad to examining the issue of

whisker orientation on the toughness of ceramic composites. The model results indicate that

crack-tip shielding due to modulus defect interactions occurs when the reinforcement has a

higher modulus than _iv matrix material, and that anti-shielding occurs for the opposite

case. Results for a single whisker located at the crack-tip (maximum effect) indicate that the

crack-tip stress intensity is reduced by about 10% when a modulus ratio of four is

assumed. Calculations performed with whisker arrays demonstrate pronounced effects of

whisker orientation on the crack-tip screening, being larger for whiskers oriented

perpendicular to the crack plane, as expectS. Ordered whisker arrays produce larger and

more uniform screening than do random whisker arrays.

Introduction

Most single phase ceramics that have useful high temperature properties suffer from a lack

of low temperature ductility and toughness. Multiphase microstructures can have better

toughness depending on the relative properties of the phases, their geometric arrangement

and the strength of their interfaces. Among the many possibilities, the addition of strong,

high modulus fibers or whiskers to brittle materials has, in many cases, yielded attractive

improvements in toughness.

Several types of toughening mechanisms in such fiber-reinforced materials have been

, advanced to explain the improvements. One example is crack bridging[I-5] that occurs

when a crack in the brittle matrix grows beyond one or more whiskers that are oriented

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MASiER

more or less perpendicular to the crack plane. Mechanically, the system can be regarded as

a kind of Dugdale zone, in which intact whiskers are replaced by a corresponding set of

pinching forces[4] acting to close the crack and, therefore, diminish the crack tip stress

intensity. This toughening effect is analogous to that created by ligament formation during

fracture of metals[6]. If the whisker/matrix interface is strong, the bridging whiskers

become highly stressed where they span the crack opening. Sliding along the

whisker/matrix interface helps smooth the stress distribution somewhat and absorbs

energy. This allows the whiskers to survive larger crack openings and to exert larger

toughening forces. Another mechanism that has been suggested is crack deflection[7,8] in

which the crack path becomes tortuous as the crack moves around whiskers and along the

interfaces.

As short fibers, whiskers can also influence toughness if they have elastic properties

different from the matrix, in which case they become sources of internal stress (the so-

called modulus defect) when an external stress is applied. Sources of internal stress, as, for

example, dislocations, microeracks, and dilatant centers, will, in general, affect the

toughness of a material via the influence they exert on the crack tip driving force [9-12]. If,

in the presence of a crack, the defect (source of internal stress) acts to decrease K, the effect

is referred to as screening or shielding, and the apparent toughness is increased because the

applied K (or equivalently the applied loading) has to be increased to obtain crack extension

[9,10]. Loss of apparent toughness is caused by antiscreening. The potential contribution

to the toughness from whiskers as modulus defects has apparently not been examined in

detail.

In the present paper, we present a a numerical procedure that approximately models the

influence of one or more whiskers on the crack tip field for an otherwise elastic solid. The

method we use is similar to that employed by Hoagland and Embury[13] who used it to

examine the effects of microeracking and by Bowling, et al. [14] in a study of toughening

contributions from both microcracks and dilatant centers. Because we retain the discrete

character of the defects, this procedure may appear to contrast with continuum approaches

like that of Budiansky, et al. [15] but, in fact, the latter is a bound of the former in the case

when the scale permits the smearing of the defects into a continuous distribution of

infinitesimal defects. On the other hand, a discretized picture incorporates a measure of the

point-to-point variability in the material and the consequences of different sequences of

, discrete events.

r

Computational Method

A great deal of computational convenience is gained by flu-stconsidering the model as two-

dimensional as shown in Figure 1.The crack length and other in plane dimensions are

regarded as much larger than the whisker size and spacing. The 2-D assumption is

reasonable if, in the thickness direction, the slab is also much larger than the whisker size

and spacing, and the whiskers in this 2-D model can then be regarded as through-the-

thickness averages of a 3-D population.

We have also approximated the 2-D shear lag stresses for an elastically dissimilar whisker

in an infinite medium to further simplify the problem. Figure 2 compares schematically

three distributions, a shear lag theory prediction [16], a stick-sUp distribution in which

sliding occurs at a constant shear stress [16-],and the linear approximation to the shear

stresses at the whisker/matrix interface used in this study. The shear lag theory has a shear

stress that varies along the length of the whisker as sinh(nx/r) where x is the distance along

the whisker axis with the origin at the midlength position. To obtain the stresses

surrounding a whisker subject to a linear shear stress distribution in an otherwise infinite

homogeneous elastic medium, we consider a Green's function for a segment of the force

acting paraUel to and along the whisker axis as represented in Figure 3. If position along a

whisker relative to its midpoint is t and the strength of the force (per unit length) acting

along the whisker is S, the stresses in the surroundings due to the forces acting on segmentdt at t are

Sdt {( y-t 4x2(y-t) }dO'xx-2Tr(_+l) K-1)i2+2y t+r 2 .(t2+2y t+r2) 2

Sdt { (._) y-t _ 4x2(y-t)2)2 } (1)d(:/YY=2Tr (_ +1) t2+2yt + r 2 (t22yt + r

Sdt {+3)t x _ 4x 3 }dO'xY=2rr(K+l) K 22y t+r 2 (t22y t+r2) 2

3

i

c

where _ = 3-4v and r2 = x2 + y2 and x and y are as defined in Figure 3. The distribution is

made linear by setting S = Wt, where q/is defined below. Integrating over the length of the

whisker from -P to +9 gives the net stresses at a x,y-field point in the infinite 2D space due

to a whisker of length 2_?at the origin. Some brevity is provided by using nondimensional

coordinates, i.e., by replacing x by x_?and y by yJL giving for the stresses

Oxy=2TrO<,+l) (K+3)[In + x ] +2[ 'r21_ " 21_ + x- e2)

(2)

where R12 = 1 - 2y + r2, R22 = 1 + 2y + r2, 81 = tan'l((1-y)/x), 82 = -tan'l((l+y)/x), _ is

half the whisker length, and it is to be understood that x, y, r and Ri are nondimensional.

The magnitude of the stresses in the matrix surrounding the whisker is controlled by the

coefficient q,, which, in turn, must be determined by the response of the system (whisker

and matrix) to an applied stress parallel to the whisker axis. In what follows we have

neglected interaction with shear components. The magnitude of this response can be

determined in the following way. Consider the force supported by the whisker

F = o J'CryydX = w.oa (_<+3) (3)(K+I)-_2

where a is the whisker diameter. Using the condition that a uniform normal stress applied at

±y = oo in the matrix, ctm, produces a stress in the whisker of c_f and that the strain in ",he

, matrix approximately matches that of the whisker, we can set F/Pa = crf - 0'm and gives for

q,,

(K+1) (4)= -1) (,<+3)

where Ef and Em are the Young's modulus of the whisker and matrix, respectively. Thus,

the strength of the stresses generated by the whisker are linearly related to the nominal

stress applied to it, the modulus ratio, and inversely to the whisker length.

Equation 2 provides the means for accommodating the presence of an arbitrarily oriented

whisker by superposing these stresses everywhere, at least at those locations of interest

such as other whiskers and along the crack faces, after transforming the stresses to account

for the rotated coordinates. The conditions involved in the derivation of equations 2 and 4,

strictly confine equation 2 to a 2D whisker immersed in a uniform stress. A further

simplification we have adopted is to neglect variations in applied stress over the length of

the whisker, and set the component of the normal stress acting parallel to the whisker axis

at the whisker mid point equal to _ m. In view of the other approximations, this assumption

probably has little substantive influence on the results except for whiskers which may be

subject to stresses that vary drastically over/heir length. Therefore, this analysis is most

appropriate to short whiskers.

The situation in which more than one whisker exists and a crack is also present is treated

numerically in the manner described in more detail in references 13 and 14. Briefly, each

whisker experiences the crack tip stress field, the magnitude of which is determined by the

two stress intensities, K1 and K2, and the stress fields of ali other whiskers in the model.

The traction free surfaces which define the crack are maintained by applying point forces

along closely spaced points on the crack faces as needed to remove the tracrSons created by

the sum of the stresses, from equation 2, of ali the whiskers. These point forces change the

nominal stress which each whisker experiences, and, in addition, change the stress

intensities. If, in the crack coordinates, Cryyand crxy are the tractions produced by the

whiskers, and 3c is an increment of length along the crack face located at the ith point a

distance ci from the crack tip, the change in mode 1 and 2 stress intensity factors are

KI="-"_/2 TrCi

(5)

AK2 .-.,_2.rrci

where the sums are carried out to a distance sufficiendy far from the tip that the tractions

have decayed to reasonably small levels. In fact the range of the stress field of a whisker is

rather short, being on the order of the whisker length, and therefore for the models

discussed here which contained up to a few hundred whiskers in a region up to 20 whisker

lengths in extent, we found that the tractions became quite small at distances of greater than

about 10 whisker distances from the crack tip.

The calculation iteratively cycles over all points, whiskers and crack faces, using the stress

increment at each point in a cycle as the Green's function for the next iteration cycle. When

the largest change in the stresses at any point becomes less than a preset value, the

calculation is considered to have converged to a solution that identifies the total stress acting

on each whisker and produces a reasonably stress free crack. Typically, we find that the

convergence is rapid, normally about three iterations to produce changes which are less

than 0.1% of the stress at a point.

In the following section, we present results that examine some of the effects produced by

one whisker, as well as the consequence of moving a crack through an array of whiskers

whose position and orientation were determined by a random number generator. Because

of the approximations discussed above, these results should be regarded as qualitative and

as representing trends.

Results

Single Whisker

We present flu,stsome results demonstrating the influence of a single whisker on the crack

! tip stress field. The magnitude of that interaction is a function of three coordinates, r, B,

and _, shown in Figure 1, where the f'u'st two define the position of the midpoint of the

whisker in crack tip coordinates, and the last, the orientation of the whisker axis relative to

the x-axis, the crack propagation direction. In Figures 4 - 6 are shown the computed

dependence of the changes in the two stress intensities, AK1 and AK2, on (D.In these

results, since the magnitude of the effects scale with the applied K, the dependence is

expressed in terms of the relative change, AKi/(woK1), where K1 is the applied stress

intensity and qJ0 is the group of elastic constants multiplying crm. The applied mode 2

stress intensity was zero. In these figures, the whisker was located at a radial distance of 2

and at several values of 9

These results (Figures 4 - 6) indicate that, as 8 increases from 0, the mode I contribution

evolves from screening for ali whisker orientations, to a more complex behavior involving

some orientations that cause screening and others that are antiscreening at 8 = 11"/2.For a

whisker with a modulus ratio of 5, the maximum screening occurs ahead of the crack tip

corresponding to a 10% decrease in K1. Also, the mode 2 contribution at most positions isof the same order as the mode 1.

One way to gain an appreciation for the spatial dependence of screening is to assume that

the number of whiskers is very large and that their orientations are uniformly distributed.

An effective, or average, screening contrit,ution can be obtained which removes the

dependence by integrating over _ from 0 t,'Jrr anddividing by Tr. The resulting average

value of the relative change in the mode 1 K is given by

11'

-7-'= _ _Kd_ (6)0

The 0-dependence of the average computed by numerical integration for the r = 2 results is

shown in Figure 7. In terms of mode 1, Figure 7 shows that whiskers tend to screen for

0<0<93 and antiscreen at 93<0<153. However, there is a net screening effect (AK1/K10

< 0) at this distance from the crack tip, for ali values of 8. Also, while we display the

results for O> 153 in Fig. 7, the assumption of matching strains in the matrix and whisker

does not apply in this range because of the discontinuity in displacements across the crack

faces and the fact that, for some qb,the whiskers straddle the crack, ha these cases, the

pinching forces mentioned above would provide a better description of the effects of thewhiskers.

Figure 8 displays examples of the radial dependence of the average mode 1 contribution for

several values of 8. The rather rapid decrease in the effect of the whisker with increasing

distance form the crack tip is evident in this figure, as is the very weak interaction that

occurs with the crack for whiskers located directly above or below the crack. The strongest

interaction occurs for whiskers ahead of the crack where: the direction of the effect is

screening.

Whisker Arrays

By monitoring the mode 1 and 2 stress intensities as the crack tip is extended in small

increments through an array of whiskers, the consequences of various multiple-whisker

arrangements on toughness can be explored. Such calculations are conducted iteratively as

described above. In the results that follow, the aspect ratio (29/a) was chosen to be 100 and

none of the whiskers were allowed to cross the crack plane. Therefore, the results show

effects solely due to modulus defect interactions. Also, ali of the whiskers have the same

W0 ( > 0). Whiskers for which -1 < W0 < 0, would produce similar results except that the

changes in the stress intensity would have opposite signs; i.e., screening would become

antiscreening and vice versa. t

Figure 9 shows results for a crack extending from x -- -6 to x = +6 through a randomly

positioned array of whiskers. There are 100 whiskers involved in this array, which, for an

aspect ratio of 100 and the size of the model used, corresponds to a rather modest volume

fraction of 4%. Three sets of calculations are presented here. In one case, the net change in

mode 1 stress intensity for randomly oriented whiskers (produced from a uniformly

distributed set of random numbers) is shown and for most, but not all, of the segment of

extension, the effect is screening, i.e. AK1 < 0. The other two sets of results are for non-

random arrays in which the whiskers have a preferred orientation produced by a normally

distributed set of random numbers with a standard deviation of Zr/10 and a mean of 11"/2in

one case (perpendicular orientation) and zero (parallel orientation) in the other. For this

particular whisker array there is a distinct advantage of a perpendicular orientation. In fact,

over a significant portion of the crack extension distance in the parallel orientation, there

exists antiscreening, although a part of this antiscreening is an artifact of the model, as will

be described below. A similar set of results for another randomly positioned array

: containing 200 whiskers but with the same volume fraction is shown in Figure 10.

When the whisker array is spatially non-random there emerges a stronger indication of the

advantage of whisker alignment, as shown in Figure 11. Here we show the results of a

crack which has been extended through a square array of perfectly oriented whiskers,

oriented perpendicular to the crack plane in one case and parallel in the other. Screening by

the perpendicular array is considerably greater. A stress intensity which oscillates with the

period of the array spacing is also evident.

A distinct trend of increasing K with crack length is also apparent in Figure 11. This trend

derives from the fact that, in this and the previous models, the whiskers are contained in a

finite size region and that the crack tip field, therefore, senses the lack of whiskers beyond

the boundaries of the model. In other words, for a short crack length there are a

preponderance of whiskers which lie in the strong screening zone ahead of the crack while

as the crack advances more whiskers enter the zone of mixed interaction. This effect also

demonstrates that, while the range of the effect exerted by a whisker decreases fairly

quickly with distance from the crack tip, there are consequence_ of ignoring the tiny

contributions from an infinite region. Indeed, if the size of the model were increased

without limit this trend would disappear. While we have not done so here, we could correct

these results by including an additional contribution to the local K that would remove this

trend by treating the material outside the model as containing a continuum of infinitesimal

whiskers and integrating over that space. This correction is discussed in more detail by

Bowling et al. [14].

Discussion

A numerical approach has been presented that lends itself to modeling the screening or

antiscreening effects due solely to modulus differences of a d_.scretearray of whiskers in an

elastic matrix. We have applied the method to single whiskers, and to examining the issue

of whisker orientation on toughness. The model results indicate that crack-tlp shielding due

solely to modulus defect interactions occurs when the reinforcement has a higher modulus

than the matrix material, and that anti-shielding occurs for the opposite case. Results for a

single whisker located at the crack-tip (maximum effect) indicate that the crack-tip stress

intensity is reduced by about 10% when a modulus ratio of four is assumed. Calculations

performed with whisker arrays demonstrate pronounced effects of whisker orientation on

the crack-tip screening, being larger for whiskers oric,nted perpendicular to the crack plane,

as expected. Ordered whisker arrays produce larger and more uniform screening than do

random whisker arrays.

There are many additional aspects of the problem which we have not addressed including

crack bridging, interface slip/failure, whisker fracture and others, However, the effects

which these additional aspects of the problem would exert could be accommodated

relatively easily. F(,r example, a critical stress criterion could be invoked that would either

limit the strength of the whisker, i.e., its qJ0, or to remove it, e.g., to raimic complete

failure of the interface, or to change its lengm simulating whisker fracture. Thus, this kind

of approach creates the opportunity for exploring a wide range of problems and

microstructures.

To compare the toughening contributions of whiskers treated as modulus defects with other

ways they may affect toughness, such as by bridging the crack, requires some

consideration of the several factors of size, aspect ratio, and modulus ratio, as well as

distribution. For modulus ratios of four, an aspect ratio of 100, and a volume fraction of

4%, these results suggest that toughness increases of about 10% above that of the matrix

should result. This is perhaps not as strong a screening effect as would derive from an

equivalent volume fraction of bridging whiskers, but it is important to recognize that many

whiskers that may bridge a crack are not oriented perpendicular to the crack plane and

therefore will be subject to large bending moments in addition to tension created that resists

the crack opening displacements. Such unfavorably aligned whiskers would not be able to

play a significant bridging role.

Acknowledgments

The authors would like to acknowledge the useful discussions and encouragement by Dr.

R. H. Jones and the discussions with Sh'of. J. P. Hirth. One of us, R. G. H., would also

like to gratefully acknowledge the support of the Pacific Northwest Laboratory in this

work. This work was supported by the Office of Basic Energy Sciences, Division of

Materials Sciences, U.S. Department of Energy under Contract DE-AC06-76RLO 1830.

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any of theiremployees, mal, es any warranty, express or implied, or assumes any legal liability or responsi-

bility for the accuracy, comple,',.eness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Refer-

ence herein to any specific commercial product, process, or service by trade name, trademark,

manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-

mendation, or favoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of the

ULited States Government or any agency thereof.

10

References

1. B. Budiansky, J. W. Hutchinson, and A. G. Evans, "Matrix Fracture in Fiber-Reinforced Ceramics", J. Mech. Phys. Solids, 34, 167-189 (1986).

2. M. Ruhle, B. J. Dalgleish, and A. G. Evans, "On the Toughening of Ceramics byWhiskers", Scripta Met., 21, 681-686 (1987).

3. D. B. Marshall and A. G. Evans, "The Tensile Strength of Uniaxially ReinforcedCeramic Fiber Composites", Fracture of Ceramics, Vol. 7, cd. by R. C. Bradt, et al.,Plenum, N.Y., 1-15 (1986).

4. A. G. Evans and R. M. McMeeking, "On the Toughening of Ceramics by StrongReinforcements", Acta Metall., 34, 2435-2441 (1986).

5. D. B. Marshall, B. N. Cox, and A. G. Evans, "The Mechanics of Matrix Cracking inBrittle-Matrix Fiber Composites", Acta MetaU., 33, 2013-2021 (19S5).

6. R.. G. Hoagland, A. R. Rosenfield, and G. T. Hahn, "Mechanisms of Fast Fracture andArrest in Steels", Met. Trans, 3, 123-136 (1972).

7. K. T. Faber and A. G. Evans, "Crack Deflection Processes-I. Theory", Acta MetaU.,31,565-576 (1983).

8. P. F. Becher, T. N. Tiegs, J. C. Ogle, and W. H. Warwick, "Toughening of Ceramicsby Whisker Reinforcement", Fracture of Ceramics,Vol. 7, cd. by R. C. Bradt, et al.,Plenum, N.Y., 61-73 (1986).

9. R. Thomson and J. E. Sinclair, "Mechanics of Cracks Screened by Dislocations", ActaMetall., 30, 1325 (1982).

10. R. Thomson, "Brittle Fracture in a Ductile Material with Application to HydrogenEmbrittlement", J. Mater. Sci., 13, 128-142 (1978).

11. J. P. Hirth, R. G. Hoagland, and C. H. Popelar, "On Virtual Thermodynamic Forceson Defects and Cracks", Acta Metall., 32, 371-379 (1984).

i2. R: G. Hoagland and J. P. Hirth, "Dilatant Center-Crack Tip Interaction Effects", J.Am. Cer. Soc., 69, 530-533 (1986).

13. R. G. Hoagland and J. D. Embu."y, "A Treatment of Inelastic Deformation Around aCrack Tip due to Microcracking", J. Am. Cer. Soc, 63, 404-410 (1980).

14. G. D. Bowling, K. T. Faber, and R. G. Hoagland, "Computer Simulations of R-Curve Behavior in Microcracking Materials", J. Am. Cer. Soc., 70, 849-854 (1987).

15. B. Budiansky, J. W. Hutchinson, and J. C. Lambropoulos, "Continuum Theory of12"latant Transformation Toughening in Ceramics", Int. J. Solids Struct., 19, 337-355(1983).

16. M. R. Piggott, Load Bearing Fibre Composites, Pergamon, N.Y., 83-99 (1980).

11

Y

////Figure 1. Schematic of the model configuration.

12

iz

Figure 2. Schematic shear stresses at whisker/matrix interface from shear lag thez_ry(curve) and the assumed linear stress distribution used in model describezi hem. Thedash line represents another alternative distribution in which sliding occurs at theinterface at a constant shear stress [16].

, Figure 3. The action of the whisker on the matrix is represented by a distribution offorces of strength S so that the segment dt contains force of magnitude Sdt. Theother symbols are defined in the text.

13

9O

12O 6O

240 300

270

Figure 4. A polar plot showing the relative change in the stress intensities,

AK1/(w0K1) (square symbols) and L_K2/(qJ0K1) (circular symbols) as a function

of the whisker axis orientation, _ Negative and positive values are represented by

open and closed symbols, respectively. The whisker midpoint is at located at r = 2,

and 0 =0.

14

90

12O 60

150 30

B

[]

210 330

240 300

270

Figure 5. Same as Figure 4 except 8 = Tr/4.

90

12O 6O

150 30

180 0

210 330

, 240 300

270

Figure 6. Same as Figure 4 except 8 = 11"/2.

15

90

120 60

150 30

180 0.02 0.02 0

210 330

240 300

270

Figure 7. A polar plot of the average mode 1 contribution, AK1/K1, as a function

of the position angle, 8, for r = 2. Again, open points are negative and closedpoints, positive. Results in the range 153<0<206 (between the dashed lines) donot reflect the fact that the whisker intersects the crack for some dp in this range.

16

0.02

. "_• _____..______._0.00

: _ _ __----1", _;_t-0.02 .......................

_° .o._ i ..

-_ .o.o_ ......-_ 90

. _ _- 135

-0.08 ....................r= ...._......................................._.....................

-0.10 ' ' , , , , •0 1 2 3 4 5 6 7

r

Figure 8. Radial dependence of the average mode 1 contribution, A K1/K1, for

several values of the position angle, 0, for a single whisker.

17

5 .'1'- _ -I- O.O8++ -I- _ -!- -t-+++++

-!- .!..!..i-I- -!-+ + + + - 0.04

+++ + + _ + ++ ++ + + 2,_

0 0.00 v

+ + + _,t,"'-

+++ ++ <1+ +

- -0.(34+ +

++ ++ + + -_

+ + + +-i-I+ + + ++++ + + . -o.o_-5 ' + '

-10 0 10

×

Figure 9. In this figure a crack was extended from left to right, starting at x = -6 tox = +6, in an array of whiskers located at +. The resulting change in mode 1 K,

expressed as AK1/(w0K1) are shown for the case where the whisker orientationsin the array are uniformly random (O), where the orientations are normallydistributed around a mean of zero (_), and normally distributed around a mean of_:/2(0).

18

Figure 10.A situation similar to that in Figure 9 except: 1)the whisker locations aredifferent, 2)the crack was extended from -6 to +4, and 3) the model is twice aslarge, and therefore contains twice as many whiskers, as that in Figure 9.

19

-' 0 0 1010 = 0.04+ +++; +++ +++ +++_ +++ +

+++++++++++++++++++ 003

++++++++++++__ +++ 002

++++++++_,_w_++++++ oo_ _....._f.+......... _°

:_, 0 -0.00•.!- -I- -I- _* -!- -f- -!- -!- -I- "1- "1-'i- "t" "t- "i" -t" "1-•

_Z

oo+ -I- -F + + + + + + + + + + -I-

. . . . . .+.+__v . . . . . . oo_+ + + _%"+ ¥ + + + + + + + + + oo3

v _

-i- -i--!--I- -I--!--I--I- +-I- + +.F-I- +-F +-I- +-10 , I i ; -0.04

-lO 0 10

X

Figure 11. The results of extending a crack through a perfectly arranged rectangular

array of whiskers. The whiskers are also perfectly oriented with the (•) results

from whiskers parallel to the crack plane and (0) perpendicular to the crack plane.

20